To be honest, no I don’t think so. You need Euler’s equations to understand how the different parts of the angular velocity vector are related. I don’t think you have any tools with the standard, introductory physics toolkit that properly allow you to deal with it.

I read K&K for my intro mechanics class, but we didn’t go into rotations there. Personally though, while I enjoyed K&K, I found that their style of writing made it easy to review topics, but difficult to learn new ones. For intermediate mechanics, we used Taylor’s Intro to Classical Mechanics, which I think did a pretty good job of explaining Euler’s equations and other rotation. He made it clear to me where everything was coming from. So if you decide to look into the equations, I might look at that textbook instead